The present invention is related at least generally to flywheel driven power storage systems and, more particularly, to an advanced flywheel driven power storage system and associated method that exhibits improved levitation force.
A flywheel may be regarded as a mechanical device, having a significant moment of inertia that may be used as a storage device for rotational energy. For example, flywheels resist changes in their rotational speed such that an energy input or an energy output is required order to change the rotational speed of a given rotating flywheel. In purely mechanical systems, such as an automobile engine, this requirement may cause the flywheel to exert a stabilizing influence on the system. The same holds for electro-mechanical systems. For example, an electrical motor may include a flywheel for resisting sudden changes in rotational speed of the motor.
In recent decades, flywheels have been employed in electro-mechanical systems that may be configured as electrical storage devices such that the devices can be electrically charged and discharged in a manner that is at least generally analogous to the charging and discharging of conventional rechargeable batteries. In this context, a flywheel can be “charged”, for storing energy by using electrical energy provided, for example, through electrical cables to increase the speed of the flywheel to cause an increase in the rotational energy. The flywheel can be subsequently discharged by decreasing the speed to cause a decrease in the rotational energy of the flywheel while converting this rotational energy into electrical energy for delivery, for example through electrical cables, from the flywheel to some external load.
An electrical machine can be attached to a rotating flywheel, for example, using a single shaft, and configured for operating in one of several modes including but not limited to (i) a charging mode with the electrical machine operating as a motor for receiving electrical energy and for increasing the rotational speed of the flywheel, and (ii) a discharging mode with the electrical machine operating as a generator for providing electrical energy via a decrease in the speed of the flywheel and (iii) a float mode with the electrical machine spinning freely without adding or subtracting rotational energy to or from the wheel. It is noted that in the float mode a given flywheel may exhibit residual power losses, for example due to frictional losses, that may cause at least gradual decreases in stored rotational energy even if no electrical power is entering or leaving the flywheel through the electrical machine. For purposes of descriptive clarity, and based on well known conventions, it is noted that an electrical machine attached to a flywheel and configured for operation in one or more of these three modes may be referred to hereinafter as a motor.
It will be readily appreciated by a person of ordinary skill in the art that a maximum energy storage capacity of a flywheel system may depend on a maximum rotational speed at which the flywheel can be rotated, without bursting or otherwise failing as a result of internal stresses for example due to centrifugal forces, and that an increase in the maximum speed causes a corresponding increase in the maximum storage capacity in proportion to the square of the maximum speed. It will be further appreciated that the maximum rotational speed can be sufficiently high that the presence of any ambient gas at atmospheric pressure can cause severe power loss and overheating that can result in catastrophic failures. For example, flywheel systems described throughout this disclosure may operate at rotational speeds well above 10,000 rpm, and, in some applications, a vacuum pressure of less than 10 mTorr is required in order to avoid excessive power loss and rotor heating. In view of these considerations and based on well known techniques, it is often necessary to contain a flywheel in a housing that at least provides an airtight seal for supporting low pressure vacuum surrounding the flywheel rotor.
Turning now to the figures, FIG. 1 is a diagrammatic elevational view, in cross-section, illustrating one example of a prior art electro-mechanical flywheel unit, generally indicated by the reference number 100, that can be utilized for storing rotational energy as part of an electrical energy storage system. Flywheel unit 100 includes a flywheel rotor assembly, generally indicated by reference number 105 having a rotatable shaft 110 that supports a rim 115 using a hub 120 for rotation as indicated by an arrow 122 in a selected direction which can be either clockwise or counterclockwise. The flywheel unit of the present example may be contained in an airtight sealed housing 125 that at least supports low vacuum therein and that provides at least some degree of containment in the event of a high speed failure such as delamination of the rim. Furthermore, flywheel housing 125 may be configured to provide structural support for a number of components therein, as will be described immediately hereinafter.
Flywheel unit 100 includes an electrical motor 126, shown within a dashed rectangle, having a motor rotor assembly 135 that is connected with shaft 110 for co-rotation with the shaft, and a motor stator assembly 140 that is supported by housing 125 through a support structure 145, as needed.
Based on well known techniques, the flywheel rotor may be radially constrained using a lower bearing assembly 150 and an upper bearing assembly 150′. The bearing assemblies can be supported by lower and upper mechanical damper assemblies 155 and 155′, respectively, that are connected to housing 125 through support structure 145.
Based on well known principles of mechanics, the maximum storage capacity of flywheel unit 100 may depend in part on the weight of the rotor such that a heavier rotor, with a given maximum speed, may exhibit a proportionally higher storage capacity as compared to a lighter rotor. Furthermore, it will be appreciated that bearing assemblies capable of operating at high maximum rotational speeds, for example above 15,000 rpm, may be incapable, at least when operating at such high speeds, of withstanding axial forces of more than just a few pounds. In this regard, it is often desirable to configure a given flywheel unit with a flywheel levitation apparatus for limiting axial forces on the bearings by supporting at least a majority of the weight of the rotor in the axial direction using a flywheel levitation apparatus.
In view of the foregoing discussions, flywheel unit 100 further includes a magnetic flywheel levitation apparatus 160, indicated in FIG. 1 within a dashed rectangle. An annular rotor 163 forms part of the levitation apparatus and is connected with shaft 110 for co-rotation and vertical movement therewith. Rotor 163 may be composed of a magnetic material such as, for example, iron. An annular stator assembly forms another part of the levitation apparatus and is indicated by the reference number 166. Stator assembly 166 is supported by support structure 145 and is configured as an electromagnet, as will be described in detail at appropriate points hereinafter, for receiving a variable electrical current (not shown) and for magnetically applying an axial lifting force upon rotor 163 that is produced, at least in part, by this current.
The flywheel levitation apparatus may be configured, as will be described in detail hereinafter, for providing sufficient upward force upon rotor 163 for fully levitating the entire weight of the flywheel rotor. In many cases, this may be regarded as a highly desirable mode of operation at least for the reason that the bearings in this mode may be subjected, at least on average, to limited axial forces, for example resulting from slight flexure of the bearing dampers, that are sufficiently low at least for avoiding short-term damage to the bearings. One well known approach for enabling full levitation of a flywheel includes configuring the upper and lower dampers to exhibit sufficient axial compliance to allow the flywheel to move vertically with correspondingly low axial restraint such that a vertical height of the flywheel rotor is primarily influenced by the flywheel levitation apparatus, and not by the dampers. It will be further appreciated that in the context of many well known fully-levitated flywheel configurations there is often a need to employ some form of closed-loop feedback, for example utilizing a position sensor, for monitoring the height of the flywheel and controllably adjusting the variable current for the lift apparatus, responsive to the vertical height, for controlling vertical, movement of the flywheel rotor assembly to at least approximately maintain some predetermined axial height.
With ongoing reference to FIG. 1, flywheel unit 100 may include a position sensor 170 for monitoring a vertical height of the flywheel rotor by measuring a sensor gap 173 having a sensor gap size that varies responsive to a vertical movement 174 indicated in FIG. 1 by a double-ended arrow. Based on well known techniques, during continuous operation of the flywheel unit, the current provided to the stator of the levitation apparatus may be varied in response to signals produced by position sensor 170, such that a predetermined sensor gap size is maintained, at least to within a range of values.
Attention is now directed to FIG. 2 which is a diagrammatic enlarged fragmentary view, in perspective, of a portion of flywheel unit 100 of FIG. 1. FIG. 2 illustrates a number features already shown in FIG. 1, and is included for purposes of clarity at least to illustrate the cylindrical symmetry of flywheel levitation apparatus 160 as well as a number of components, including rotor 163 and stator assembly 166.
Attention is now directed to FIG. 3, which is a diagrammatic cutaway view, in elevation, of a portion of flywheel unit 100 including flywheel levitation apparatus 160 for purposes of facilitating the introduction of further details with respect to its operation. As described previously, rotor 163 is connected for co-rotation with shaft 110 and may be composed of a magnetic material such as, for example, iron. Stator assembly 166 is supported by support structure 145 and is configured to receive a variable electrical current. The stator assembly generates a total magnetic flux 176, schematically indicated in FIG. 3 by a solid line, that may include a variable magnetic flux 178, produced by an electrical coil 180 responsive to the variable current, and a steady magnetic flux 182 produced, for example, by a permanent magnet 184. It should be appreciated that a single permanent magnet may be used, or a suitable arrangement of a plurality of permanent magnets. For this reason, item 184 may be referred to below as a permanent magnet arrangement that can be made up of one or more permanent magnets. It is noted that the fluxes are shown in the components to the left of shaft 110 in the view of the figure in order to facilitate illustrative clarity, but are understood to be present throughout the annular configuration of these components as is the case for magnetic flux illustrations in subsequent ones of the figures. In the illustrated embodiment, electrical coil 180 may be supported in a number of different ways, and may be wound based on well known techniques commonly employed, for example, in production of conventional voice coils or other magnetic actuators. It is noted that, in certain cases, the flywheel may be operated at sufficiently low vacuum pressure such that the immediate ambient surroundings near the coil provide little or no heat path to reduce heating caused by resistive losses in the coil, and it may be desirable to support the electrical coil in a manner that provides an adequate heat path for maintaining a sufficiently low temperature in the coil. For example, it may be desirable to encase the coil within a thermally conductive resin 185 such as a thermally conductive two-part epoxy. An inner pole piece 186 includes a projecting section 188 that extends downward toward an upper surface 190 of rotor 163. The inner pole piece is configured, based on well known principles of magnetism, for channeling the magnetic flux toward upper surface 190 of the rotor, and across a rotor-stator gap 192 to exert a magnetic lifting force 194 upon the rotor, as indicated in FIG. 3 by an arrow. Again, the reference numbers applied to components of the stator assembly have been shown only to the right of shaft 110 for purposes of clarity, but it is to be understood that these components are annular in configuration.
The stator assembly further includes a permanent magnet arrangement having permanent magnet 184 arranged in magnetic communication with inner pole piece 186 for contributing to the magnetic flux. An outer pole piece 196 is arranged in magnetic communication with permanent magnet 184 and extends downward toward upper surface 190 of rotor 163. In one conventional embodiment, the permanent magnet arrangement may include a single ring-shaped magnet having (i) a north pole N that is aligned in a confronting relationship with an input surface of the inner pole piece and (ii) a south pole S that confronts an additional input surface of the outer pole piece. The magnet poles are not required to be oriented as shown and may be reversed. It will be readily appreciated by a person of ordinary skill in the art that the inner pole piece and outer pole piece of the present embodiment are configured to cooperate with one another, and with the magnet and the rotor, such that total magnetic flux 176 is channeled through a magnetic circuit as indicated in FIG. 3, that passes through (i) the outer pole piece, (ii) the rotor and (iii) the inner pole piece. It will be further appreciated that total magnetic flux 176 includes one contribution from the permanent magnet, which produces steady magnetic flux 182 to cause a corresponding steady upward force 204 on the rotor, and another contribution from the coil, based on variable magnetic flux 178 to produce a variable upward force 206. The latter varies responsive to a variable current provided to the coil, for example, from a control unit (not shown). It is noted that total upward force 194 produced by total magnetic flux 176 includes the sum of steady upward force 204 and variable upward force 206. Further, it should be appreciated that the steady force is steady for a given width of gap size 173 but can change in magnitude responsive to changes in the gap size, as anyone who has handled a permanent magnet will appreciate. In the instance of a permanent magnet arrangement including a plurality of permanent magnets that cooperate with one another to define a pair of opposing magnet surfaces diagrammatically shown by reference numbers 208a and 208b using dashed lines, the pole piece(s) provide an input surface confronting each magnet surface for receiving the steady magnetic flux. The pole pieces are aligned in a confronting relationship to receive the steady magnetic flux for channeling the flux to the rotor. The steady magnetic flux, produced by the magnet array, can exhibit spatial fluctuations across each input surface. A pole piece and/or pole pieces in cooperation can serve to smooth the steady magnetic flux such that the lower surface of the pole pieces exhibit output fluctuations that are reduced as compared to the spatial fluctuations at each input surface 208a and 208b. 
While the foregoing description embraces sufficient details for at least supporting a general qualitative understanding of an overall manner in which the total magnetic field is channeled, a person of ordinary skill in the art will recognize that accurate quantitative analysis may require a more detailed three dimensional computational determination of the magnetic fields in and around the levitation apparatus. For example, with respect to a given embodiment of the levitation apparatus, it may be advantageous to provide accurate computations using finite element analysis (FEA) to determine a magnitude of steady force 204.
Attention is now directed to FIG. 4, which is a magnetic field map, generally indicated by the reference number 210 that has been produced by Applicant based on well known techniques in conjunction with commercially available software for performing finite element analysis (FEA). As indicated in FIG. 4 by the appropriate respective reference numbers, inner pole piece 186, outer pole piece 196, permanent magnet arrangement 184 and rotor 163 are diagrammatically represented, in cross-section, in a manner that is consistent with FIGS. 1-3. Magnetic field lines 216 (one of which is indicated) are superposed on these diagrams to represent a set of computational data corresponding to a three dimensional distribution of the total magnetic field, in and around the levitation apparatus, caused solely by the permanent magnet arrangement. The computational data has been produced using commercially available ANSYS® FEA software (in conjunction with a commonly available desktop personal computer) based in part on the geometrical shapes of these components, and their approximate magnetic material properties, as manually entered by Applicant through the standard ANSYS® graphical user interface. It is noted that while the components at hand are three dimensional objects, all being of annular configuration as explicitly illustrated by FIG. 2, it nevertheless will be appreciated by a person of ordinary skill in the art that the cylindrical symmetry of these shapes causes the magnetic fields, and therefore the computational data representative thereof, to exhibit rotational symmetry about a central axis of symmetry (not shown) such that it is sufficient, at least for purposes of visualization, to display the results in two dimensions. It is further noted that at least in the context of these computations, previously described coil 180 and resin 185 can be assumed to be composed of substantially non-magnetic material (such as copper, epoxy, and polymeric wire insulation) and therefore can be expected to cause no influence, at least to within an approximation, on magnetic fields in and around the levitation apparatus. Therefore, for purposes of illustrative clarity, the coil and the epoxy are omitted in FIG. 4.
A person of ordinary skill in the art, being familiar with these standard computational techniques, will recognize that at any given point, at least within conventional magnetic materials such as iron, the magnetic field at any given location (or point) in space may be at least approximately quantitatively characterized according to (i) an intensity and (ii) a spatial orientation. Based on well known conventions for graphically representing magnetic fields, the spatial orientation of the field at a location 214 (or any other location) can be represented by a field line 216 such that each field line represents a particular orientation. The intensity of the corresponding flux at that location can be interpreted as being inversely proportional to spacing between adjacent ones of the field lines, by way of example, in the manner of showing elevation on a topographic map.
As will be discussed in greater detail at appropriate points hereinafter, a magnetic field map such as that of FIG. 4 may provide a person of ordinary skill in the art at least with substantive qualitative insights as to the expected performance of a given magnetic system. Furthermore, inspection of field maps such as that of FIG. 4 may provide a person of ordinary skill in the art with useful insights for interpreting computational results as opposed to attempting to interpret a relatively massive amount of raw computational data. Concerning the latter point, it is noted that computational data produced by ANSYS® (or its equivalent) may be manipulated for reliably qualitatively characterizing various characteristics of the magnetic system with a high degree of accuracy. For example, as will be described immediately hereinafter, it is readily possible using the ANSYS software, and by employing well-known techniques, to process the computational data, represented by FIG. 4, for determining a magnitude of steady force 204 (FIG. 3) between stator 166 and rotor 163, corresponding to a gap size for rotator-stator gap 192, and to repeat the calculation for a range of gap sizes to provide a plot of the upward force for different gap sizes.
It is again noted that the field lines illustrated in FIG. 4 have been computed based solely on contributions of the permanent magnet arrangement, and therefore represent only the steady magnetic field. These field lines do not explicitly include any contribution from the variable magnetic field provided through the coil responsive to the variable magnetic current. It is to be understood, however, that the pole pieces will tend to channel the variable magnetic field, produced by the variable current through the coil, in substantially the same way that the pole pieces channel the steady magnetic field produced the permanent magnet, causing the magnetic field lines at least in and around rotor-stator gap 192 to exhibit at least approximately similar spatial distributions of directional orientations, for the variable and for the steady magnetic fields, even while the respective magnitudes of the variable and steady magnetic fields may be very different.
Attention is now turned to FIG. 5, which is a plot, generally indicated by the reference number 219, having a vertical axis 220, corresponding to a magnitude of steady force 204 due to the permanent magnet arrangement, and a horizontal axis corresponding to the value of gap 192. As described immediately above, Applicant employed well known techniques, in conjunction with ANSYS® software to determine a magnetic force profile 222 representing the magnitude of steady force 204, for example due to the permanent magnet arrangement, for a number of different gap sizes. In particular, for each gap size, the ANSYS® software was utilized for producing a corresponding set of computational data representing the magnetic field in and around the levitation apparatus, and for each set of computational data, a subset of that set was processed, through ANSYS, employing well known techniques for computing the force associated with that gap size.
Considering FIGS. 1 and 3 in conjunction with FIG. 5, it is noted that the magnitude of force 204, per plot 222, increases rapidly as the gap size diminishes, and that curve becomes increasingly nonlinear as the gap size approaches 40 mils. Applicant recognizes herein that this rapid nonlinear increase in force 204 may introduce substantial problems and/or challenges with respect to configuring and operating the flywheel levitation unit, as will be described at appropriate points hereinafter.
Referring once again to FIG. 3, Applicant recognizes that at least in the context of the illustrated flywheel unit, one design goal may be to limit the vertical movement of the flywheel rotor to at least generally avoid contact between the rotor and stator of the flywheel levitation apparatus. One well known technique for achieving this is to constrain or restrict the vertical movement of the flywheel rotor to a predetermined maximum height such that the rotor-stator gap is unable to decrease below a corresponding minimum value. In one embodiment, sensor 170 may be encased in a rigid sensor housing 224 defining a lower surface region 226, that faces an uppermost portion 228 of the shaft and is positioned to act as a stop that constrains vertical movement of the flywheel rotor to a predetermined maximum height. For example, surface region 226 may be aligned to prevent the rotor stator gap from decreasing below 40 mils.
Applicant further recognizes that the vertical movement of the flywheel, as described immediately above, may be limited in a way that at least generally avoids uncontrolled upward vertical movement of the flywheel. In particular, Applicant recognizes that if the flywheel lift apparatus is to be configured for controllably levitating the entire amount of rotated weight for a range of values of rotor-stator gap 192, it may at least be of benefit that steady upward force 204 (from the permanent magnet) can by itself be incapable of lifting the entire weight of the flywheel rotor, at least throughout the total, range of values for gap 192. It is further noted that if this is not the case, and if the steady upward force can lift the entire weight of the flywheel, then vertical movement of the flywheel within the total range of values of the gap size, could cause the steady upward force to initiate an uncontrolled upward vertical movement that may cause the permanent magnet to lift the flywheel rotor assembly and continuously maintain the rotor at a maximum height, for example with uppermost portion 228 pressed firmly against surface region 226.
Turning to FIG. 1, based on well known practice, a touch-down arrangement 230 may be arranged in a spaced apart relationship with a lowermost portion 232 of the flywheel shaft and configured for limiting downward movement of the flywheel such that a sufficient downward movement causes the lower portion of the shaft to be received by the touch-down arrangement at a height corresponding to a maximum allowed value of the rotor-stator gap.
As noted above, with reference to FIG. 5, the rapid nonlinear increase in force responsive to decreasing gap size may introduce substantial problems and/or challenges with respect to configuring and operating the flywheel levitation apparatus, as will be described immediately hereinafter with reference to one particular example.
As one particular example, a flywheel levitation apparatus having the force profile 222 of FIG. 5 may be configured, by employing the techniques described above, such that a minimum gap size is approximately 40 mils. With this restriction in place, it is evident based on FIG. 5 that a 4000 lb. flywheel rotor assembly may be levitated in a way that at least generally avoids uncontrolled upward vertical movement due solely to the steady magnetic field. Furthermore, touchdown arrangement 230 may be configured, in accordance with the foregoing description, for limiting the downward vertical movement such that the maximum size of gap 192 is 140 mils. In this configuration, the flywheel unit is operable through a total range of gap sizes corresponding to a total allowed range of 100 mils of vertical movement. While it may be desirable to controllably levitate the flywheel rotor assembly to a particular height within this range, for example corresponding to a gap size of 90 mils, and to controllably maintain a predetermined rotor-stator gap size, at least to within a predetermined range of values that deviates from the predetermined value by, for example, substantially less than 50 mils in either the upward or downward direction, it will be appreciated by a person of ordinary skill in the art that there could be a number of reasons to provide for a total allowed range of vertical movement that exceeds the predetermined range associated with routine operation. For example, in the context of a particular flywheel unit, the given total range of movement of 100 mils may be required in order to account for considerations including but not limited (i) thermal expansion of the flywheel rotor assembly, (ii) flexure of the flywheel rotor during operation, (iii) manufacturing assembly tolerances of the rotor as well as the support structure of the flywheel unit and (iv) shock and vibration during normal operation. As will be described in greater detail hereinafter, the last of these considerations may be of particular significance, since disturbances in the form of shock and vibration tend to be inevitable under typical conditions of operation.
As noted above with reference to FIG. 5, Applicant appreciates that the rapid nonlinear increase in steady force 204 may introduce substantial problems and/or challenges with respect to configuring and operating the flywheel levitation apparatus, as will be described immediately hereinafter.
Referring to FIGS. 1 and 3, in conjunction with FIG. 5, it is evident that with a minimum gap size of 40 mils, and a total range of vertical movement of 100 mils, the maximum gap size will be at least approximately 140 mils. Based on FIG. 5, for a gap size of 140 mils, steady upward force 204 due to the steady magnetic field, produced by the permanent magnet arrangement, is approximately 1800 lbs and is thus substantially less than the entire weight of the flywheel rotor assembly. It is therefore evident that in order to fully levitate the 4000 pound rotor off of the touchdown bearings, for example during initial start up of the flywheel unit, or in the event that a disturbance causes a touchdown of the flywheel rotor assembly, it may be necessary to provide sufficient variable current through coil 180 to cause the variable magnetic field to produce over 2000 pounds of force as upward variable force 206. Furthermore, Applicant appreciates that if the coil is unable to contribute over 2000 pounds of upward variable force, then during the course of normal operation a given disturbance, such as an externally induced vibration of the flywheel housing, may cause the flywheel to drop in an uncontrolled downward vertical movement whereupon the coil is unable to controllably prevent the drop.
Applicants appreciate that it may be challenging, at least for a number of practical reasons, to configure the stator of the levitation apparatus for providing a variable magnetic field of such a magnitude as just described. In this regard, there are a number of practical challenges associated with configuring a flywheel levitation apparatus for producing a high variable upward force. As one example of a practical challenge, the coil may be required to receive a high current for producing thousands of pounds of lifting force. Resulting resistive losses may produce overheating unless some form of external cooling is applied to the coil. As another example, configurations that satisfy excessive requirements for variable upward lifting force may require correspondingly excessive physical bulk in the levitation apparatus, possibly including a large radius. Applicants appreciate that this latter concern may be of paramount significance at least for the reason that radial stresses in a rotor of large radius may be excessive at high speeds, and there may be a need to limit the outer diameter of the rotor. Based at least on the foregoing points, a person of ordinary skill in the art will readily appreciate that a requirement for a large upward variable force may introduce substantial challenges with regard to configuring a levitation apparatus for levitating a given flywheel.
Summarizing with respect to the above example, a conventional flywheel levitation apparatus may be configured to provide a steady magnetic flux 182, produced by a permanent magnet, to produce steady force 204 in accordance with the plot of FIG. 5, and vertical movement of the flywheel apparatus may be limited to a predetermined total range of vertical movement (for example 100 mils) corresponding to a minimum gap size (for example 40 mils) and a maximum gap size (for example 140 mils). While current may be applied to the coil to levitate a heavy flywheel at some predetermined gap size, the rapid nonlinear increase in force 204 may lead to the aforedescribed requirement that the stator provide a variable electromagnetic field capable of lifting a significant fraction of the total weight of the flywheel rotor at least for certain positions in the total allowed vertical movement of the flywheel rotor. Applicants appreciate that it may be impractical to provide a levitation apparatus capable of providing a variable magnetic field for lifting a large portion of the total weight of a flywheel, at least for the reasons that a levitation apparatus satisfying these requirements may require an excessively large overall size including a large rotor plate, and for a given configuration the coil may be excessively prone to overheating. Based at least on the foregoing concerns which Applicant now recognizes with respect to the prior art, Applicant further recognizes that the prior art has failed to recognize, much less resolve the concerns and related problems that have been brought to light with respect to a flywheel levitation apparatus that overcomes these challenges.
The foregoing examples of the related art and limitations related therewith are intended to be illustrative and not exclusive. Other limitations of the related art will become apparent to those of skill in the art upon a reading of the specification and a study of the drawings.